Liu, Zeqing; Agarwal, Ravi P.; Feng, Chi; Kang, Shin Min Weak and strong convergence theorems of common fixed points for a pair of nonexpansive and asymptotically nonexpansive mappings. (English) Zbl 1099.47057 Acta Univ. Palacki. Olomuc., Fac. Rerum Nat., Math. 44, 83-96 (2005). Let \(K\) be a subset of a Banach space. A mapping \(T:K\to K\) is said to be asymptotically nonexpansive if there exists a sequence \(\{k_n\}\) in \([1,\infty)\) such that \(\lim_n k_n=1\) and \(\| T^nx-T^ny\| \leq k_n\| x-y\| \) for all \(n\in\mathbb{N}\) and for all \(x,y\in K\). In particular, if \(k_n\equiv1\), we call \(T\) a nonexpansive mapping. In this paper, the authors introduce a new three-step iteration method with error terms for a pair of nonexpansive and asymptotically nonexpansive mappings. They obtain some strong and weak convergence theorems for the iteration whenever the space is uniformly convex. An example is included to show that their result substantially extends the corresponding results of S. S.Chang [Indian J.Pure Appl. Math. 32, No. 9, 1297–1307 (2001; Zbl 1034.47039)], Z.–Q.Liu and S. M.Kang [Acta Math.Sin., Engl.Ser.20, No. 6, 1009–1018 (2004; Zbl 1098.47059)] and M. O.Osilike and S. C.Aniagbosor [Math.Comput.Modelling 32, No. 10, 1181–1191 (2000; Zbl 0971.47038)]. Reviewer: Satit Saejung (Khon Kaen) Cited in 3 Documents MSC: 47J25 Iterative procedures involving nonlinear operators 47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc. 47H10 Fixed-point theorems Keywords:nonexpansive mapping; asymptotically nonexpansive mapping; fixed point Citations:Zbl 1034.47039; Zbl 0971.47038; Zbl 1098.47059 PDFBibTeX XMLCite \textit{Z. Liu} et al., Acta Univ. Palacki. Olomuc., Fac. Rerum Nat., Math. 44, 83--96 (2005; Zbl 1099.47057) Full Text: EuDML References: [1] Chang S. S.: On the approximation problem of fixed points for asymptotically nonexpansive mappings. Indian J. Pure Appl. Math. 32 (2001), 1297-1307. · Zbl 1034.47039 [2] Chang S. S.: Some problems and results in the study of nonlinear analysis. Nonlinear Anal. TMA 30 (1997), 4197-4208. · Zbl 0901.47036 · doi:10.1016/S0362-546X(97)00388-X [3] Goebel K., Kirk W. A.: A fixed point theorem for asymptotically nonexpansive mappings. Proc. Amer. Math. Soc. 35 (1972), 171-174. · Zbl 0256.47045 · doi:10.2307/2038462 [4] Gornicki J.: Weak convergence theorems for asymptotically nonexpansive mappings in uniformly convex Banach spaces. Comment. Math. Univ. Carolina 30 (1989), 249-252. · Zbl 0686.47045 [5] Liu Z., Kang S. M.: Weak and strong convergence for fixed points of asymptotically nonexpansive mappings. Acta. Math. Sinica 20 (2004), 1009-1018. · Zbl 1098.47059 · doi:10.1007/s10114-004-0321-7 [6] Opial Z.: Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull. Amer. Math. Soc. 73 (1967), 591-597. · Zbl 0179.19902 · doi:10.1090/S0002-9904-1967-11761-0 [7] Osilike M. O., Aniagbosor S. C.: Weak and strong convergence theorems for fixed points of asymptotically nonexpansive mappings. Math. Comput. Modelling 32 (2000), 1181-1191. · Zbl 0971.47038 · doi:10.1016/S0895-7177(00)00199-0 [8] Rhoades B. E.: Fixed point iteration for certain nonlinear mappings. J. Math. Anal. Appl. 183 (1994), 118-120. · Zbl 0807.47045 · doi:10.1006/jmaa.1994.1135 [9] Schu J.: Iterative construction of fixed points of asymptotically nonexpansive mappings. J. Math. Anal. Appl. 158 (1991), 407-413. · Zbl 0734.47036 · doi:10.1016/0022-247X(91)90245-U [10] Schu J.: Weak and strong convergence of fixed points of asymptotically nonexpansive mappings. Bull. Austral. Math. Soc. 43 (1991), 153-159. · Zbl 0709.47051 · doi:10.1017/S0004972700028884 [11] Xu H. K.: Inequalities in Banach spaces with applications. Nonlinear Anal. TMA 16 (1991), 1127-1138. · Zbl 0757.46033 · doi:10.1016/0362-546X(91)90200-K This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.